PHY331 Magnetism

Lecture 1

Overview •  Course syllabus / general information •  Quick revision of basic concepts •  Magnetization and susceptibility •  Using susceptibility to define magnetic

materials – Diamagnetic – Paramagnetic – Ferromagnetic

•  Summary

•  Lecture 1: General introduction and revision, dipoles, magnetic materials, magnetisation, susceptibility.

•  Lecture 2: Magnetic dipole moment of a circulating electron. •  Lecture 3: Langevin’s theory of diamagnetism. •  Lecture 4: Classical treatment of paramagnetic susceptibility. •  Lecture 5: Magnetic dipole moment of an atom via Hund’s Rules. •  Lecture 6: Quantum theory of paramagnetism. •  Lecture 7: Domain theory of ferromagnetism. Antiferromagnets. •  Lecture 8: Spontaneous magnetisation and the exchange interaction. •  Lecture 9: Weiss molecular field model of ferromagnetism. •  Lecture 10: Paramagnetic susceptibility of free electrons (Pauli

paramagnetism).

SYLLABUS

What if we can’t understand the lecture notes?

The material is covered in the two recommended text books,

“Introduction to Solid State Physics” Charles Kittel 7th Edition (John Wiley & sons)

Chapters 14 and 15

“Solid State Physics” J. R. Hook & H. E. Hall 2nd Edition (John Wiley & sons) Chapters 7 and 8

All of these notes can be downloaded from PHY331 website. Can also get .pdf versions of the notes. These contain a little more ‘background’ information.

www.sheffield.ac.uk/physics/teaching/phy331/index.htm

Magnets - what’s the big attraction?

•  i) important physical state and •  ii) of considerable technological

significance (all electrical motors and transformers,

magnetic fields for all purposes, including medical, magnetic storage, sensors, security tags, etc etc)

Magnetic fields from conduction currents (i)

Biot Savart Law

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dB = µ0idLsinθ4πr2

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dB = µ0i2πrThe strength of the magnetic interaction

is defined by µ0 which is known as the permeability of free space. µ0 has a value of 4 x 10-7 Hm-1. The unit of B is the tesla (T).

For an infinitely long wire

See lecture 10 of 2nd year EM notes

Force on a current carrying element Experimentally the magnetic force dF acting on a current element length dL carrying a current i and placed in a uniform field B is found to be

The direction of the magnetic force is normal to the plane containing both B and dL. In vector notation:

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dF = BidLsinθ

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dF = idL × B

Magnetic ‘dipoles’

•  Easiest way to think of a magnetic dipole is as a result of a current flowing in a miniature wire. Leads naturally to a picture of electron ‘currents’ in atoms.

•  This results in a magnetic dipole moment m, defined by a current i, and a vector area A. Arrow shows the sense of the vector area.

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m = iAA i m

Magnetic fields… Magnetic induction field (B-field)

Potential (scalar) field

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B = −∇VM

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VM =mcosθ4πr2

θ

N S

Placing a magnetic dipole in a B field •  The energy U of a magnetic dipole m in a uniform Magnetic

Induction Field B

•  The torque Γ on a magnetic dipole m in a uniform Magnetic Induction Field B (Torque is a measure of how much a force acting on an object causes that object to rotate)

•  Scalar field can be used to ‘generate’ forces (which are usually vector fields). In general, the force F can be described by the gradient of a scalar field U, i.e.

•  The force F on a magnetic dipole m in a non-uniform Magnetic Induction Field B

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U = − m ⋅B

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Γ = m×B

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F = − ∇m⋅B€

F = − ∇U

B-field and H-field •  Can view a magnetic material as being composed of many

individual current-carrying loops - each with a magnetic dipole moment. If all loops are identical, then current flow in the material is zero.

•  However the effects of the magnetic dipoles can be modelled by thinking of them resulting from a surface current termed an Amperian current (see L15, 2nd year EM course).

•  Also have magnetic fields that result from the flow of ‘real’ conduction currents.

•  Both currents (Amperian and conduction) can contribute to the B-field. However only conduction currents can contribute to the H-field.

•  Can write B field in terms of the Magnetization of the material and the conduction currents that flow.

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B = µo(H + M)From magnetic material From conduction currents

Magnetization •  Each small volume dτ of a magnetized material will

posses a magnetic dipole moment dm.

•  Magnetization is defined as the magnetic dipole moment per unit volume M = dm / dτ (units Am-1)

The magnetisation of materials •  In the presence of a magnetic material, there will

be two contributions to the total Magnetic Induction Field B

Using our relation between B, H and M

We define the susceptibility (chi) as χ = M / H

so that,

and define, so that,

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B = Bcurrent elements +Bmagnetic materials

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B = µ0H +µ0M

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B = µ0H +µ0χH

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B = µ0 1+χ( )H

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µr = 1+χ( )

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B = µrµ0H

•  Here µr is the relative permeability of the material, which we use in place of, µ0 the permeability of free space.

•  All the equations used when there are no magnetic materials are simply modified by replacing,

µ0 with µrµ0

when magnetic materials are present.

Units •  When M and H both have the (same) units of

amperes / meter, then susceptability (χ) is called the “volume magnetic susceptability” and is dimensionless.

•  There are however two other (SI) measures of susceptibility, the mass magnetic susceptibility (χmass), measured in m3 kg−1 and the molar magnetic susceptibility (χmol) measured in m3mol−1

•  Can convert between these using ρ the density in kg m−3 and M (molar mass) kg mol−1.

χmass = χ / ρ

χmol = Mχmass = Mχ / ρ

How do we classify magnetic materials?

Depending on χ, we class all materials as being

Diamagnetic, Paramagnetic or Ferromagnetic.

Diamagnetic materials

Examples χ (per kg)

bismuth -1.7 x 10-8 copper -0.107 x 10-8 germanium -0.15 x 10-8 gold -0.19 x 10-8 hydrogen -2.49 x 10-8

helium -0.59 x 10-8

Discuss diamagnetism lecture 2 / 3

χ < 0, i.e negative and µr < 1 small negative magnetisation.

Diamagnetic levitation of a frog in a magnetic field

Paramagnets Characterized by χ > 0 and µr > 1

Examples χ × 10-6 (per kg) aluminium 0.82 calcium 1.40 magnesium 0.69 platinum 1.65 tantalum 1.10

Discuss paramagnetism lecture 4 / 6 / 10

Ferromagnets

χ > 0, and µr >> 1 Large positive magnetisation

•  Examples χ between 102 - 103 but only an in ‘initial χ’ is it is proportional to H

•  Examples Iron, nickel, cobalt, NiFe, FeCo alloys etc and other amorphous alloys

•  Discuss ferromagnetism in lectures 7 / 8 / 9

Summary

•  Revised basic concepts (B and H-field, energy, torque and force in a magnetic field).

•  Introduced magnetization, susceptability and relative permeability.

•  Talked about different types of magnetic materials (diamagnetic, paramagnetic, ferromagnetic).